Question

Example: A coin is tossed twice. Let X denote the number of head on the first toss and Y denote the total number of heads on the 2tosses. Construct the join probability mass function of X and Y is given below and answer the following questions.

f(x, y)		x		Row Total
		0	1
y	0
	1
	2
Column Total

(a) Find P(X = 0,Y <= 1)

(b) Find P(X + Y = 2)

(c) Find P(Y ≤ 1)

(d) Find the conditional probability P (X = 0 | Y ≤ 1)

Answer 1

f(x,y)		x		Row toatl
		0	1
y	0	1/4	0	1/4
	1	1/4	1/4	1/2
	2	0	1/4	1/4
column total		1/2	1/2

as there are four possible outcomes,

$\left \{ HH(X=1,Y=2) ,HT(X=1,Y=1),TH(X=0,Y=1),TT(X=0,Y=0)\right \}$

note that,

$P(X=0,Y=2)=0=P(X=1,Y=0)$

a)

$P(X=0,Y\leq 1)$

$=P(X=0,Y=0)+P(X=0,Y=1)$

$=1/4+1/4$

$=1/2$

b)

$P(X+Y=2)=P(Y=2-X)$

$when, \quad X=0,Y=2\quad and \quad when \quad X=1,Y=1$

$\implies P(X+Y=2)=P(X=0,Y=0)+P(X=0,Y=1)$

$=0+1/4$

$=1/4$

c)

$P(Y\leq 1)$

$=P(Y=0,X=0,1)+P(Y=1,X=0,1)$

$=1/4+1/2$

$=3/4$

d)

$P(X=0|Y\leq 1 )$

$=\frac{P(X=0,Y\leq 1 )}{P(Y\leq 1)}$

$=\frac{1/2}{3/4}$

$=\frac{2}{3}$